I think nearly every liberal (to include all of their alt accounts) I have encountered on the site have an issue coping with reality.
Being said; in answer to your question; no, I do not think all liberals are woke.
There are liberals that have not imbibed the leftist kool aid
The liberals that have resisted are seeking dialogue to gain compromise on social issues.
The liberals that succumbed to the left are incited by imagined victimhood to agitate against "the man" when in fact they are agitating against themselves.
Example: The gays and lesbians had won their place in society after years of blood, sweat, mockery, murder and tears.
Along comes the liberal left with the "trans" fad nonsense and messed things up. They got people looking at the LGB sideways again.
Then add the "eat the rich", "free palestine", "blm", "antifa", "illegal immigrants", "homeless crisis", garbage and America became a dumpster.
To me it seems the liberal leftist puppets are fast asleep having vivid dreams of conquering injustices that do not exist on the way to creating a great socialist utopia.
Then, having proven themselves useful and obedient, they will be bestowed with the great honor and glory of becoming slaves of the state.
P.S. "woke" was stolen and from black people and adulterated.
Leftist liberal "karens" need to return the term to its rightful owners.